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Y commutes. Then Y E is called an exponential object. Exponentials may not exist for a given pair of objects Y; E, but if an exponential Y E does exist, it is unique up to isomorphism. Exercise: suppose C is a category with pullbacks, exponentials, and a nal object 1. X Z Y ' Exercise: suppose that exponentials always exist in C. Given an arrow X ! Y, E show that there is a unique arrow ' making the diagram "Y Y Y E E ????! x 'E 1E ? XE x? ' X E ????! "X commute. Verify that this de nes a functor, called the exponential functor, from C to itself, together with a natural transformation from the exponential functor onto the identity functor.

Bell, Toposes and Local Set Theories. Oxford: Clarendon Press, 1988. 3. Garrett Birko and Saunders Mac Lane, A Survey of Modern Algebra. Fourth edition. New York: Macmillan, 1977. 4.

Axiom of choice holds Every epic is a retraction Figure 5. The nature of the mathematics founded upon a given topos depends on the structure of that topos. 1Y . ) But this is easily restated very succintly in categorical terms: the Axiom of Choice holds in a topos T i every epic is a retraction. Moreover, Diaconescu showed that this happens only if T is a Boolean topos; see 1] 2] 17]. 3. Let T be a topos. A Lawvere-Tierney operator on T is an arrow j : ! such that: 1. j > = >, 2. j j = j , 3.

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